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in K and to each prime ideal P lying over p in K there is a unique element g of Gal(K/Q) satisfying the condition g(x) = x^p (mod P) for all integers x of K. Varying P over p changes g into a conjugate (and every conjugate of g occurs in this way), so the conjugacy class of g in the Galois group is canonically associated to p. This is called the Frobenius conjugacy class of p and any element of the conjugacy class is called a Frobenius element of p. If we take for K the mth cyclotomic field,

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